1. The problem is to understand how to perform algebraic long division, which is a method to divide one polynomial by another. 2. The formula used is similar to numerical long division but applied to polynomials: divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient. 3. Important rules: - Always arrange polynomials in descending powers. - Subtract the product of the divisor and the current quotient term from the dividend. - Bring down the next term and repeat until the degree of the remainder is less than the divisor. 4. Example: Divide $$x^3 + 2x^2 - 5x + 6$$ by $$x - 1$$. 5. Step 1: Divide leading terms: $$\frac{x^3}{x} = x^2$$. 6. Step 2: Multiply divisor by $$x^2$$: $$(x - 1)(x^2) = x^3 - x^2$$. 7. Step 3: Subtract: $$\left(x^3 + 2x^2 - 5x + 6\right) - \left(x^3 - x^2\right) = x^3 + 2x^2 - 5x + 6 - x^3 + x^2 = 3x^2 - 5x + 6$$. 8. Step 4: Divide leading terms: $$\frac{3x^2}{x} = 3x$$. 9. Step 5: Multiply divisor by $$3x$$: $$(x - 1)(3x) = 3x^2 - 3x$$. 10. Step 6: Subtract: $$\left(3x^2 - 5x + 6\right) - \left(3x^2 - 3x\right) = 3x^2 - 5x + 6 - 3x^2 + 3x = -2x + 6$$. 11. Step 7: Divide leading terms: $$\frac{-2x}{x} = -2$$. 12. Step 8: Multiply divisor by $$-2$$: $$(x - 1)(-2) = -2x + 2$$. 13. Step 9: Subtract: $$\left(-2x + 6\right) - \left(-2x + 2\right) = -2x + 6 + 2x - 2 = 4$$. 14. Since the remainder $$4$$ has degree less than the divisor $$x - 1$$, the division stops. 15. Final answer: Quotient is $$x^2 + 3x - 2$$ and remainder is $$4$$, so $$\frac{x^3 + 2x^2 - 5x + 6}{x - 1} = x^2 + 3x - 2 + \frac{4}{x - 1}$$.